Most of the open problems in computational complexity can be stated in
terms of lower bounds. There are few general techniques
available in proving lower bounds, and consequently, there has been
little progress in this field. My approach to solving such problems is
bottom up, working with simple open problems such as:

Do we need an exponential number of states to simulate a two-way
nondeterministic finite automaton by a deterministic one?

What is the minimum parallel time required to solve some
specific problems using a given number of processors?